10 Basic Parent Functions and Transformation Rules

This set of flashcards covers the 10 basic parent functions—including linear, quadratic, exponential, and logarithmic forms—listing their mathematical properties like domain, range, symmetry, and end behavior, along with fundamental algebraic transformation rules.

Linear Parent Function ($y=x$)

An odd function with domain $(-\infty, \infty)$ and range $(-\infty, \infty)$. End behavior: as $x \rightarrow -\infty, y \rightarrow -\infty$ and as $x \rightarrow \infty, y \rightarrow \infty$. Critical points: $(-1, -1), (0, 0), (1, 1)$.

Absolute Value Parent Function ($y=|x|$)

An even function with domain $(-\infty, \infty)$ and range $[0, \infty)$. End behavior: as $x \rightarrow -\infty, y \rightarrow \infty$ and as $x \rightarrow \infty, y \rightarrow \infty$. Critical points: $(-1, 1), (0, 0), (1, 1)$.

Quadratic Parent Function ($y=x^2$)

An even function with domain $(-\infty, \infty)$ and range $[0, \infty)$. End behavior: as $x \rightarrow -\infty, y \rightarrow \infty$ and as $x \rightarrow \infty, y \rightarrow \infty$. Critical points: $(-1, 1), (0, 0), (1, 1)$.

Radical (Square Root) Parent Function ($y=\sqrt{x}$)

A function categorized as neither even nor odd with domain $[0, \infty)$ and range $[0, \infty)$. End behavior: as $x \rightarrow 0, y \rightarrow 0$ and as $x \rightarrow \infty, y \rightarrow \infty$. Critical points: $(0, 0), (1, 1), (4, 2)$.

Cubic Parent Function ($y=x^3$)

An odd function with domain $(-\infty, \infty)$ and range $(-\infty, \infty)$. End behavior: as $x \rightarrow -\infty, y \rightarrow -\infty$ and as $x \rightarrow \infty, y \rightarrow \infty$. Critical points: $(-1, -1), (0, 0), (1, 1)$.

Cube Root Parent Function ($y=\sqrt[3]{x}$)

An odd function with domain $(-\infty, \infty)$ and range $(-\infty, \infty)$. End behavior: as $x \rightarrow -\infty, y \rightarrow -\infty$ and as $x \rightarrow \infty, y \rightarrow \infty$. Critical points: $(-1, -1), (0, 0), (1, 1)$.

Exponential Parent Function ($y=b^x, b>1$)

A function categorized as neither even nor odd with domain $(-\infty, \infty)$ and range $(0, \infty)$. It has a horizontal asymptote at $y=0$ and critical points at $(-1, \frac{1}{b}), (0, 1), (1, b)$.

Logarithmic Parent Function ($y=\log_b(x), b>1$)

A function categorized as neither even nor odd with domain $(0, \infty)$ and range $(-\infty, \infty)$. It has a vertical asymptote at $x=0$ and critical points at $(\frac{1}{b}, -1), (1, 0), (b, 1)$.

Rational (Inverse) Parent Function ($y=\frac{1}{x}$)

An odd function with domain $(-\infty, 0) \cup (0, \infty)$ and range $(-\infty, 0) \cup (0, \infty)$. It has asymptotes at $y=0$ and $x=0$ and critical points at $(-1, -1), (1, 1)$.

Rational (Inverse Squared) Parent Function ($y=\frac{1}{x^2}$)

An even function with domain $(-\infty, 0) \cup (0, \infty)$ and range $(0, \infty)$. It has asymptotes at $x=0$ and $y=0$ and critical points at $(-1, 1), (1, 1)$.

Greatest Integer Function ($y=\text{int}(x)=\llbracket x \rrbracket$)

A function categorized as neither even nor odd with domain $(-\infty, \infty)$ and range $\{y: y \in \mathbb{Z}\}$ (integers). End behavior: as $x \rightarrow -\infty, y \rightarrow -\infty$ and as $x \rightarrow \infty, y \rightarrow \infty$.

Constant Parent Function ($y=C$)

An even function with domain $(-\infty, \infty)$ and range $\{y: y=C\}$. End behavior: as $x \rightarrow \pm\infty, y \rightarrow C$. Critical points: $(-1, C), (0, C), (1, C)$.

Vertical Shift

A transformation rule where $g(x) = f(x) + c$ shifts the graph up and $g(x) = f(x) - c$ shifts the graph down.

Horizontal Shift

A transformation rule where $g(x) = f(x + c)$ shifts the graph left and $g(x) = f(x - c)$ shifts the graph right.

Reflections

A transformation rule where $g(x) = -f(x)$ flips the graph over the $x$-axis and $g(x) = f(-x)$ flips the graph over the $y$-axis.

Vertical Stretch or Compression

For $g(x) = c f(x)$, the graph stretches vertically if $c > 1$ and compresses vertically if $0 < c < 1$.

Horizontal Stretch or Compression

For $g(x) = f(cx)$, the graph compresses horizontally if $c > 1$ and stretches horizontally if $0 < c < 1$.